Lemma 1

Tag 009B, permanent. Cite it as [Bourbaki, Tag 009B].

Algebra, VIII, p. 139, § 8. Semisimple Rings

Let A be a left Artinian ring, and let M be a simple A-module. Then the ring $A_M$ is simple.

Since the ring A is left Artinian, the same holds for the ring $A_M$ , by Proposition 5 of VIII, p. 7. Now, M is a faithful and simple $A_M$ -module, so the ring $A_M$ is simple (VIII, p. 119, Proposition 1).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Proposition 5, § 8
Proposition 8, § 8

Cites

Proposition 5 of VIII, p. 7
VIII, p. 119, Proposition 1